DM2 The Fermi - Pasta - Ulam - Tsingou chain

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Introduction

Consider a system which evolves deterministically (there is no noise) and observe it for a long time. You can e.g. think of hard spheres on a billiard, or of particles interacting on a lattice, or more generally of a system described by a Hamiltonian dynamics
$\dot p_n = - \frac{\partial\mathcal H}{\partial q_n}\quad$
$\dot q_n = \frac{\partial\mathcal H}{\partial p_n} \quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad(1)$

(Here, qn(t) represents a position and pn(t)a momentum). The initial condition is qn(0), pn(0). The total energy is conserved: it is constant in time:

$E=\mathcal H(q_n(0),p_n(0)) \, .$
One may hope that in the long time and large system size limits, one recovers the results of statistical physics: in other words, we expect that the long-time average of an observable coincides with the ensemble average of statistical physics. In this case, energy is conserved, the ensemble is microcanonical. For this ensemble we expect equipartition: the probability density for the system to have visited a point (qn, pn) in the phase space is uniform on the energy shell E :
$E=\mathcal H(q_n,p_n)$
One prerequisite for this property to hold is that the system has to explore the whole energy surface. There are simple counter-examples to this expectation: one is given by a chain of harmonically coupled particles on a lattice (i.e. particles connected to their neighbors with elastic springs). Roughly speaking, the number of constants of motion in this system is high enough to forbid the dynamics to explore the whole energy surface.

The harmonic chain

Fourier representation

Particle are distributed on a lattice close to integer sites 0,1,...L. Each particle interacts with its closest neighbors with an elastic potential V. We consider the case with fixed boundary conditions.

We start by studying a discrete elastic chain: L+1 particles are interacting on a one dimensional lattice (see figure). The particles interact with their neighbors at left and at right with a harmonic potential. The distance between the rest position of the particle n and its actual position is denoted by un (n=0...L). We consider the case with fixed boundary conditions and with the particles initially at rest:
$u_0=u_L=0, \quad \dot u_0=\dot u_L=0, \quad v_n(0)=0$
The Hamiltonian is
$\mathcal H(\{u_n,v_n\}) = \frac 12 \sum_{n=0}^{L-1} \Big[ v_n^2 + (u_{n+1}-u_n)^2\Big]$
where the velocity is the moment conjugated to the position (taking unit mass). Using equation (1), show that the equations of motion are:
$\ddot u_n = u_{n+1}+u_{n-1}-2u_n \quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad(2)$
The problem is easier to study in the Fourier space: one defines the Fourier modes k = 0,1...L − 1
$\hat u_k = \sqrt{\tfrac 2L}\sum_{n=0}^{L-1} u_n \sin\frac{nk\pi}{L}\quad ; \quad\quad\quad \hat v_k = \sqrt{\tfrac 2L}\sum_{n=0}^{L-1} v_n \sin\frac{nk\pi}{L}$
which are compatible with the boundary conditions. The inverse Fourier transform is given by
$u_n = \sqrt{\tfrac 2L}\sum_{k=0}^{L-1} \hat u_k \sin\frac{nk\pi}{L}\quad ; \quad\quad\quad v_n = \sqrt{\tfrac 2L}\sum_{k=0}^{L-1} \hat v_k \sin\frac{nk\pi}{L}$
The corresponding expression of the Hamiltonian is
$\mathcal H(\{\hat u_k,\hat v_k\}) = \sum_{k=0}^{L-1} E_k = \sum_{k=0}^{L-1} \frac 12 \Big[ \hat v_k^2 + \omega_k^2 \hat u_k^2\Big] \quad\text{where}\quad \omega_k=2\sin\frac{k\pi}{2L}$

• Count the modes of strictly positive energy for the system described above.
• At t=0, we prepare the system at energy E. If the equipartition of energy applies, what is the average energy associated to each mode k?

Evolution in time

At t=0, we prepare the system in the Fourier mode k=1:
$u_n(t=0) ~=~ L\sin\frac{n\pi}{L} \quad\quad v_n(t=0) ~=~ 0$

and the energy is concentrated in the first mode, of energy
$E(t=0)=E_1=L^3 \sin^2 \frac{\pi}{2L}$
We may wonder whether the energy stays in this mode or decays on the others modes, as predicted by statistical physics. We now study this question numerically, by discretizing the equation of evolution (2) with a time step dt.

import pylab,math,numpy
Ntime = 1000
L = 32
dt = 0.001
def HARMONIC(u,n):
    return u[n+1]+u[n-1]-2.*u[n]
def FT(e,k):
    return sum([e[n]*math.sin(n*k*math.pi/L) for n in range(1,L)])/math.sqrt(L/2.)
#initial condition for the position u and velocity v
u = [0]+[L*math.sin(n*math.pi/L) for n in range(1,L)]+[0]
v = [0. for n in range(L+1)]
mode1 = []
#dynamics with fixed boundary conditions: u[0] = u[L-1] = 0
for itime in range(Ntime):
    oldu = u[:]
    oldv = v[:]
    v = [0]+[oldv[n]+dt*HARMONIC(oldu,n) for n in range(1,L)]+[0]
    u = [0]+[oldu[n]+dt*v[n] for n in range(1,L)]+[0]
    k = 1; en1 = FT(v,k)**2/2.+2.*(math.sin(k*math.pi/(2*L)))**2*FT(u,k)**2
    #k = 2; en2 = FT(v,k)**2/2.+2.*(math.sin(k*math.pi/(2*L)))**2*FT(u,k)**2
    mode1.append(en1)
pylab.title('The harmonic chain')
pylab.xlabel('time')
pylab.ylabel('E_1, energy of mode 1')
time = [dt*itime for itime in range(Ntime)]
exact = [L*(L*math.sin(math.pi/(2*L)))**2 for itime in range(Ntime)]
pylab.plot(time,exact,'b-')
pylab.plot(time,mode1,'r-')
pylab.axis([0,Ntime*dt,75.,82.])
pylab.show()

Note that one could use the Fast Fourier Transform algorithm to compute The Fourier Transform. Moreover, we could also use more efficient numerical integration such as the Leapfrog method, instead of the modified Euler one.

• Check the precision of the algorithm when decreasing the time step dt.
• Try the initial condition u=[0]+[L/2.-abs(L/2.-n) for n in range(1,L)]+[0]. Plot some snapshots of the displacement u(t). Plot the time evolution of the energy Ek for modes k=1 to 4. Comment.
• Questions on Python language:

1. What is the difference between a list (e.g. [1,2,3]), a tuple (e.g. (1,2,4))?
2. Define a list a (e.g. a=[1,2,2]). Define b=a. Modify a. What happens to b?
3. If a and b are two lists, what is the difference between the commands a.append(b), a.extend(b) and a+b?

More generically, integrable systems also break ergodicity. It was thought that non-integrable systems would on the contrary be ergodic, until...

A counter-example by Fermi, Pasta, Ulam and Tsingou

System and ergodicity

In 1955, Enrico Fermi, John Pasta, Stanislaw Ulam and Mary Tsingou proposed to add a slightly non-harmonic interaction, that would allow the mixing of the energy between the modes so as to recover the microcanonical prediction of statistical physics. The equations of motions that they considered is
$\ddot u_n = u_{n+1}+u_{n-1}-2u_n + \alpha \Big[(u_{n+1}-u_n)^2-(u_{n}-u_{n-1})^2\Big] \quad \text{with} \; \alpha \ll 1$
The numerical study of this equation was performed with one of the very first computers, the Maniac I. To repeat the same experiment, you can use the following force

def FPU(u,n):
    return u[n+1]+u[n-1]-2.*u[n]+alpha*((u[n+1]-u[n])**2-(u[n]-u[n-1])**2)

instead of HARMONIC(u,n). We start with the following parameters:

Ntime = 10000
L = 32
alpha = 0.3
dt = 0.2

and this initial state

u = [0]+[math.sin(n*math.pi/L) for n in range(1,L)]+[0]

1. What do you observe? Explain why Fermi, Pasta, Ulam and Tsingou were happy.
2. Take a larger number of time steps Ntime=44000. What do you really conclude about ergodicity?
3. Compare different values of α.
4. Plot the shape of the profile un at different times on a same graph, slightly shifted so as to visualize the evolution.

Study of solitons (bonus; difficult)

This complicated evolution corresponds to the propagation of many undeforming profiles, called "solitons". To determine the possible shape of a single soliton, one first takes a macroscopic limit by defining continuous variables
$x=\varepsilon n \quad \text{and} \quad\tau=\varepsilon , \quad \text{with} \varepsilon \ll 1$

• One sets a field φ(x,τ) from the definition

$u_n(t)=\varepsilon \phi(\varepsilon n, \epsilon t)$
Show that at second order the field φ(x,τ) verifies the equation
$\partial_\tau^2\phi = \partial_x^2\phi+\bigg[2\alpha\partial_x\phi\partial_x^2\phi +\frac 1{12} \partial_x^4\phi\bigg]\varepsilon^2$

• To study a travelling wave, one sets X = x − τ. Find the equation for φ(X,τ).
• Assuming that variations in time are slow, neglect the term

$\partial_\tau^2 \phi$
and show that
$\partial_\tau\Phi +\bigg[\alpha\Phi\partial_X\Phi +\frac 1{24} \partial_X^3\phi\bigg]\varepsilon^2=0 \quad \text{where} \Phi = \partial_x \phi$

• This equation is called the Korteweg-de Vries equation. Find a traveling wave solution, i.e. of the following form:

$\Phi(X,t)= C_0 \Phi_0\big(C_1(X-v\tau)\big)$
where the invariant shape is
$\Phi_0(\bar X)=\frac 1{\cosh^2\bar X}$

• Come back to the initial variable φ(x,τ) (Hint: when integrating from Φ to φ, ensure that the profile φ goes to zero at infinity).
• Test numerically the stability of the obtained soliton. Don't forget to compute the initial velocity correctly.
• How do two colliding solitons of opposite velocities interact? Illustrate this fact numerically.

References

• The original article by Fermi, Pasta and Ulam.
• Scholarpedia article on the Fermi-Pasta-Ulam-Tsingou problem.
• Mathematical aspects of the Fermi-Pasta-Ulam-Tsingou problem.
• An historical perspective.

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